Paley–Wiener theorem

Remarks

The implication from 2 to 1 is direct: $\abs{e^{izt}} = e^{-t \operatorname{Im} z} \le e^{\sigma \abs{z}}$ for $\abs{t} \le \sigma$, and the Plancherel theorem gives the square-integrability on $\R$. For the converse, $\varphi$ is the Fourier transform of $F$ on $\R$, up to a constant. It vanishes outside $[-\sigma, \sigma]$; this follows by shifting the line of integration into the upper or lower half-plane. The growth there is controlled by the Phragmén–Lindelöf principle. For example, with $\varphi = 1/2$ on $[-\sigma, \sigma]$ one gets $F(z) = \sin(\sigma z)/z$, which has exponential type $\sigma$ and is square-integrable on $\R$. The square-integrability is needed: $\sin(\sigma z)$ has exponential type $\sigma$ but is not of the form 2, since it is not square-integrable on $\R$.

Going deeper

For a student

The space of functions in the theorem is the Paley–Wiener space, a Hilbert space of entire functions that is the basic example of a de Branges space. It is the setting of the sampling theorem of signal processing: such a function is determined by its values at the points $\pi n/\sigma$, $n \in \Z$. There are related Paley–Wiener theorems for smooth functions and distributions with compact support, where the growth condition also involves $\abs{\operatorname{Im} z}$.